Courses

UNDERGRADUATE COURSES

APMA 0090. Introduction to Modeling
Topics of Applied Mathematics, introduced in the context of practical applications where defining the problems and understanding what kinds of solutions they can have is the central issue. Computations are performed in MATLAB; instruction is provided.

APMA 0160. Introduction to Computing Sciences
For students in any discipline that may involve numerical computations. Includes instruction for programming in MATLAB. Applications include solution of linear equations (with vectors and matrices) and nonlinear equations (by bisection, iteration, and Newton's method), interpolation, and curve-fitting, difference equations, iterated maps, numerical differentiation and integration, and differential equations. Prerequisite: MATH 0100 or its equivalent.

APMA 0330, 0340. Methods of Applied Mathematics I,II
Mathematical techniques involving differential equations used in the analysis of physical, biological and economic phenomena. Emphasis on the use of established methods, rather than rigorous foundations. I: First and second order differential equations. II: Applications of linear algebra to systems of equations; numerical methods; nonlinear problems and stability; introduction to partial differential equations; introduction to statistics. Prerequisite: MATH 0100.

APMA 0350, 0360. Methods of Applied Mathematics I,II
Follows APMA 0330, APMA 0340. Intended primarily for students who desire a rigorous development of the mathematical foundations of the methods used, for those students considering one of the applied mathematics concentrations, and for all students in the sciences who will be taking advanced courses in applied mathematics, mathematics, physics, engineering, etc. Three hours lecture and one hour recitation. MATH 0180 is desirable as a corequisite. Prerequisite: MATH 0100.

APMA 0410. Mathematical Methods in the Brain Sciences
Basic mathematical methods commonly used in the cognitive and neural sciences. Topics include: introduction to differential equations, emphasizing qualitative behavior; introduction to probability and statistics, emphasizing hypothesis testing and modern nonparametric methods; and some elementary information theory. Examples from biology, psychology, and linguistics. Prerequisites: MATH 0100 or equivalent.

APMA 0650. Essential Statistics
A first course in statistics emphasizing statistical reasoning and basic concepts. Comprehensive treatment of most commonly used statistical methods through linear regression. Elementary probability and the role of randomness. Data analysis and statistical computing using Excel. Examples and applications from the popular press and the life, social and physical sciences. No mathematical prerequisites beyond high school algebra.

APMA 1070. Quantitative Models of Biological Systems

An introduction to the use of quantitative modeling techniques in solving problems in biology. Each year one major biological area is explored in detail from a modeling perspective. The particular topic will vary from year to year. Mathematical techniques will be discussed as they arrive in the context of biological problems. Prerequisites: Introductory Level Biology, APMA 0330, APMA 0340, or APMA 0350, APMA 0360, or written permission.

APMA 1080. Inference in Genomics and Molecular Biology

Sequencing of genomes has generated a massive quantity of fundamental biological data. We focus on drawing traditional and Bayesian statistical inferences from these data, including: alignment of biopolymer sequences; prediction of their structures, regulatory signals; significances in database searches; and functional genomics. Emphasis is on inferences in the discrete high dimensional spaces. Statistical topics: Bayesian inference, estimation, hypothesis testing and false discovery rates, statistical decision theory. Prerequisite: APMA 1650 or MATH 1610; BIOL 0200 recommended; Matlab or programming experience. Enrollment limited to 20.

APMA 1170. Introduction to Computational Linear Algebra
Focuses on fundamental algorithms in computational linear algebra with relevance to all science concentrators. Basic linear algebra and matrix decompositions (Cholesky, LU, QR, etc.), round-off errors and numerical analysis of errors and convergence. Iterative methods and conjugate gradient techniques. Computation of eigenvalues and eigenvectors, and an introduction to least squares methods. A brief introduction to Matlab is given. Prerequisites: MATH 0520 is recommended, but not required.

APMA 1180. Introduction to the Numerical Solution of Partial Differential Equations
Fundamental numerical techniques for solving ordinary and partial differential equations. Overview of techniques for approximation and integration of functions. Development of multistep and multistage methods, error analysis, step-size control for ordinary differential equations. Solution of two-point boundary value problems, introduction to methods for solving linear partial differential equations. Introduction to Matlab is given but some programming experience is expected. Prerequisites: APMA 0330, 0340 or 0350, 0360. APMA 1170 is recommended.

APMA 1200. Operational Analysis: Probabilistic models
Basic probabilistic problems and methods in operations research and management science. Methods of problem formulation and solution. Markov chains, birth-death processes, stochastic service and queueing systems, the theory of sequential decisions under uncertainty, dynamic programming. Applications. Prerequisite: APMA 1650 or MATH 1610, or equivalent.

APMA 1210. Operations Research: Deterministic Methods (ENGN 1310)
An introduction to the basic mathematical ideas and computational methods of optimizing allocation of effort or resources, with or without constraints. Linear programming, network models, dynamic programming, and integer programming.

APMA 1330. Methods of Applied Mathematics III, IV
Review of vector calculus and curvilinear coordinates. Partial differential equations. Heat conduction and diffusion equations, the wave equation, Laplace and Poisson equations. Separation of variables, special functions, Fourier series and power series solution of differential equations. Sturm-Liouville problem and eigenfunction expansions.

APMA 1360. Topics in Chaotic Dynamics
Overview and introduction to dynamical systems. Local and global theory of maps. Attractors and limit sets. Lyapunov exponents and dimensions. Fractals: definition and examples. Lorentz attractor, Hamiltonian systems, homoclinic orbits and Smale horseshoe orbits. Chaos in finite dimensions and in PDEs. Can be used to fulfill the senior seminar requirement in applied mathematics. Prerequisites: Differential equations and linear algebra.

APMA 1650. Statistical Inference I
APMA 1650 begins an integrated first course in mathematical statistics. The first half of APMA 1650 covers probability and the last half is statistics, integrated with its probabilistic foundation. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, confidence intervals, and hypothesis testing. Prerequisite: MATH 0100 or its equivalent.

APMA 1660. Statistical Inference II
APMA 1660 is designed as a sequel to APMA 1650 to form one of the alternative tracks for an integrated year's course in mathematical statistics. The main topic is linear models in statistics. Specific topics include likelihood-ratio tests, nonparametric tests introduction to statistical computing, matrix approach to simple-linear and multiple regression, analysis of variance, and design of experiments. Prerequisite: APMA 1650 or equivalent, basic linear algebra.

APMA 1670. Statistical Analysis of Time Series
Time series analysis is an important branch of mathematical statistics with many applications to signal processing, econometrics, geology, etc. The course emphasizes methods for analysis in the frequency domain, in particular, estimation of the spectrum of a time-series, but time domain methods are also covered. Prerequisite: elementary probability and statistics on the level of APMA 1650-1660.

APMA 1680. Nonparametric Statistics
A systematic treatment of the distribution-free alternatives to classical statistical tests. These non-parametric tests make minimum assumptions about distributions governing the generation of observations, yet are of nearly equal power to the classical alternatives. Prerequisite: APMA 1650 or equivalent.

APMA 1690. Computational Probability and Statistics
Examination of probability theory and statistical inference from the point of view of modern computing. Random number generation, Monte Carlo methods, simulation, and other special topics. Prerequisites: calculus, linear algebra, APMA 1650, or equivalent. Some experience with programming desirable.

APMA 1700. The Mathematics of Insurance
The course consists of two parts. The first treats life contingencies, i.e., the construction of models for individual life insurance contracts. The second treats the Collective Theory of Risk, which constructs mathematical models for the insurance company and its portfolio of policies as a whole. Suitable also for students proceeding to the Institute of Actuaries examinations. Prerequisites: Probability to the level of APMA 1650 or MATH 1610.

APMA 1710. Information Theory (CSCI 1850, ENGN 1510)
Information theory is the study of the fundamental limits of information transmission and storage. This course, intended primarily for advanced undergraduates, and beginning graduate students, offers a broad introduction to information theory and its applications: Entropy and information; lossless data compression, communication in the presence of noise, capacity, channel coding; source-channel separation; lossy data compression.

APMA 1720. Monte Carlo Simulation with Applications to Finance
The course will cover the basics of Monte Carlo and its applications to financial engineering: generating random variables and simulating stochastic processes; analysis of simulated data; variance reduction techniques; binomial trees and option pricing; Black-Scholes formula; portfolio optimization; interest rate models. The course will use MATLAB as the standard simulation tool. Prerequisites: APMA 1650 or MATH 1610. Offered in alternate years.

APMA 1930, APMA 1940. Senior Seminars
Independent study and special topics seminars in various branches of applied mathematics, change from year to year. Recent topics include Mathematics of Speculation, Scientific Computation, Coding and Information Theory, Topics in Chaotic Dynamics, and Software for Mathematical Experiments.

APMA 1930A. Actuarial Mathematics
A seminar considering selected topics from two fields: (1) life contingencies-the study of the valuation of life insurance contracts; and (2) collective risk theory, which is concerned with the random process that generates claims for a portfolio of policies. Topics are chosen from Actuarial Mathematics, 2nd ed., by Bowers, Gerber, Hickman, Jones, and Nesbitt. Prerequisite: knowledge of probability theory to the level of AM 165 or MA 161. Particularly appropriate for students planning to take the examinations of the Society of Actuaries

APMA 1930B. Computational Probability and Statistics
Examination of probability theory and mathematical statistics from the perspective of computing. Topics selected from: random number generation, Monte Carlo methods, limit theorems, stochastic dependence, Bayesian networks, probabilistic grammars.

APMA 1930C. Information Theory
Information theory is the mathematical study of the fundamental limits of information transmission (or coding) and storage (or compression). This course offers a broad introduction to information theory and its real-world applications. A subset of the following is covered: entropy and information; the asymptotic equipartition property; theoretical limits of lossless data compression and practical algorithms; communication in the presence of noise-channel coding, channel capacity; source-channel separation; Gaussian channels; Lossy data, compression.

APMA 1930D. Mixing and Transport in Dynamical Systems
Mixing and transport are important in several areas of applied science, including fluid mechanics, atmospheric science, chemistry, and particle dynamics. In many cases, mixing seems highly complicated and unpredictable. We use the modern theory of dynamical systems to understand and predict mixing and transport from the differential equations describing the physical process in question. Prerequisites: AM 33,34 or AM 35,36.

APMA 1930E. Ocean Dynamics
Works through the popular book by Henry Stommel entitled A View of the Sea. Introduces the appropriate mathematics to match the physical concepts introduced in the book.

APMA1930G. The Mathematics of Sports
Topics to be discussed will range from the determination of who won the match, through biomechanics, free-fall of flexible bodies and aerodynamics, to the flight of ski jumpers and similar unnatural phenomena. Prerequisites: AM 11 and AM 34 or their equivalents, or permission of the instructor.

APMA 1930H. Scaling and Self-Similarity
The themes of scaling and self-similarity provide the simplest, and yet the most fruitful description of complicated forms in nature such as the branching of trees, the structure of human lungs, rugged natural landscapes, and turbulent fluid flows. This seminar is an investigation of some of these phenomena in a self-contained setting requiring a little more mathematical background than high school algebra. Topics to be covered: Dimensional analysis, empirical laws in biology, geosciences, and physics and the interplay between scaling and function; an introduction to fractals; social networks and the “small world” phenomenon.

APMA 1930J. Mathematics of Random Networks
An introduction to the emerging field of random networks and a glimpse of some of the latest developments. Random networks arise in a variety of applications including statistics, communications, physics, biology and social networks. They are studied using methods from a variety of disciplines ranging from probability, graph theory and statistical physics to nonlinear dynamical systems. Describes elements of these theories and shows how they can be used to gain practical insight into various aspects of these networks including their structure, design, distributed control and self-organizing properties. Prerequisites: Advanced calculus, basic knowledge of probability.

APMA 1940. Senior Seminar

APMA 1940A. Coding and Information Theory
In a host of applications, from satellite communication to compact disc technology, the storage, retrieval, and transmission of digital data relies upon the theory of coding and information for efficient and error-free performance. This course is about choosing representations that minimize the amount of data (compression) and the probability of an error in data handling (error-correcting codes). Prerequisite: A knowledge of basic probability theory at the level of AM 165 or MA 161.

APMA 1940B. Information and Coding Theory
Originally developed by C.E. Shannon in the 1940s for describing bounds on information rates across telecommunication channels, information and coding theory is now employed in a large number of disciplines for modeling and analysis of problems that are statistical in nature. This course provides a general introduction to the field. Main topics include entropy, error correcting codes, source coding, data compression. Of special interest will be the connection to problems in pattern recognition. Includes a number of projects relevant to neuroscience, cognitive and linguistic sciences, and computer vision. Prerequisites: High school algebra, calculus. MATLAB or other computer experience helpful. Prior exposure to probability theory/statistics helpful.

APMA1940C. Introduction to Mathematics of Fluids
Equations that arise from the description of fluid motion are born in physics, yet are interesting from a more mathematical point of view as well. Selected topics from fluid dynamics introduce various problems and techniques in the analysis of partial differential equations. Possible topics include stability, existence and uniqueness of solutions, variational problems, and active scalar equations. No prior knowledge of fluid dynamics is necessary.

APMA 1940D. Iterative Methods
Large, sparse systems of equations arise in many areas of mathematical application and in this course we explore the popular numerical solution techniques being used to efficiently solve these problems. Throughout the course we will study preconditioning strategies, Krylov subspace acceleration methods, and other projection methods. In particular, we will develop a working knowledge of the Conjugate Gradient and Minimum Residual (and Generalized Minimum Residual) algorithms. Multigrid and Domain Decomposition Methods will also be studied as well as parallel implementation, if time permits.

APMA 1940E. Mathematical Biology
This course is designed for undergraduate students in mathematics who have an interest in the life sciences. No biological experience is necessary, as we begin by a review of the relevant topics. We then examine a number of case studies where mathematical tools have been successfully applied to biological systems. Mathematical subjects include differential equations, topology and geometry.

APMA 1940F. Mathematics of Physical Plasmas
Plasmas can be big, as in the solar wind, or small, as in fluorescent bulbs. Both kinds are described by the same mathematics. Similar mathematics describes semiconducting materials, the movement of galaxies, and the re-entry of satellites. We consider how all of these physical systems are described by certain partial differential equations. Then we invoke the power of mathematics. The course is primarily mathematical. Prerequisites: APMA 0340 or 0360, MATH 0180 or 0200 or 0350, and PHYS 0060 or PHYS 0080 or ENGN 0510.

APMA 1940G. Multigrid Methods
Mulitgrid methods are a very active area of research in Applied Mathematics. An introduction to these techniques will expose the student to cutting-edge mathematics and perhaps pique further interest in the field of scientific computation.

APMA 1940H. Numerical Linear Algebra

This course will deal with advanced concepts in numerical linear algebra. Among the topics covered: Singular Value Decompositions (SVD) QR factorization, Conditioning and Stability and Iterative Methods.

APMA 1940I. The Mathematics of Finance
The mathematics of speculation as reflected in the securities and commodities markets. Particular emphasis placed on the evaluation of risk and its role in decision-making under uncertainty. Prerequisite: basic probability.

APMA 1940J. The Mathematics of Speculation
The course will deal with the mathematics of speculation as reflected in the securities and commodities markets. Particular emphasis will be placed on the evaluation of risk and its role in decision making under uncertainty. Prerequisite: basic probability.

APMA 1940K. Fluid Dynamics and Physical Oceanography
Introduction to fluid dynamics as applied to the mathematical modeling and simulation of ocean dynamics and near-shore processes. Oceanography topics include: overview of atmospheric and thermal forcing of the oceans, ocean circulation, effects of topography and Earth's rotation, wind-driven currents in upper ocean, coastal upwelling, the Gulf Stream, tidal flows, wave propagation, tsunamis.

APMA 1940L. Mathematical Models in Biophysics
Introduction to reaction models for biomolecules, activation and formation of macro-molecules, stochastic simulation methods such as Langevin models and Brownian dynamics. Applications to blood flow, platelet aggregation, and interactions of cells with blood vessel walls.

APMA 1940. Cryptography (MATH 1580)
Topics include symmetric ciphers, public key ciphers, complexity, digital signatures, applications and protocols. MATH 1530 is not required for this course. What is needed from abstract algebra and elementary number theory will be covered. Prerequisite: MATH 0520 or MATH 0540.

APMA 1940M. The History of Mathematics
The course will not be a systematic survey but will focus on specific topics in the history of mathematics such as Archimedes and integration, Oresme and graphing, Newton and infinitesimals, simple harmonic motion, the discovery of ‘Fourier’ series, the Monte Carlo method, reading and analyzing the original texts. A basic knowledge of calculus will be assumed.

APMA 1940N. Mathematical Models in Computational Biology.
This course is designed to introduce students to the use of mathematical models in biology as well as some more recent topics in computational biology. Mathematical techniques will involve difference equations and dynamical systems theory ordinary differential equations and some partial differential equations. These techniques will be applied in the study of many biological applications as (i) Difference equations: population dynamics, red blood cell production, population genetics; (ii) Ordinary differential equations: Predator-prey models, Lotka-Volterra model, modeling and evolution of the genome, heart beat model/cycle, transmission dynamics of HIV and gonorrhea; (iii) Partial differential equations: tumor growth, modeling evolution of the genome, pattern formation. Prerequisites: APMA 0330 and APMA 0340.

APMA 1940O. Approaches to Problem Solving in Applied Mathematics
The aim of the course is to illustrate through the examination of unsolved (but elementary) problems the ways in which professional applied Mathematicians approach the solution of such questions. Ideas considered include: choosing the “simplest” nontrivial example, generalization and specification. Ways to think outside convention. Some knowledge of probability and linear algebra helpful. Suggested reading: “How to solve it,” by G. Polya and “Nonplussed,” by Julian Havil.

APMA 1940P. Biodynamics of Block Flow and Cell Locomotion

APMA 1940Q Filtering Theory
Filtering (estimation of a "state process" from noisy data) is an important area of modern statistics. It is of central importance in navigation, signal and image processing, control theory and other areas of engineering and science. Filtering is one of the exemplary areas where the application of modern mathematics and statistics leads to substantial advances in engineering. This course will provide a student with the working knowledge sufficient for cutting edge research in the field of nonlinear filtering and its practical applications. Topics will include: hidden Markov models, Kalman and Wiener filters, optimal nonlinear filtering, elements of Ito calculus and Wiener chaos, Zakai and Kushner equations, spectral separating filters and wavelet based filters, numerical implementation of filters. We will consider numerous applications of filtering to speech recognition, analysis of financial data, target tracking and image processing. No prior knowledge in the field is required but a good understanding of the basic Probability Theory (APMA1200 or APMA2630) is important.

APMA 1970. Independent Study

GRADUATE LEVEL COURSES

APMA 2050, APMA 2060. Mathematical Methods of Applied Science Introduces science and engineering graduate students to a variety of fundamental mathematical methods. Topics include linear algebra, complex variables, Fourier-series, Fourier and Laplace transforms and their applications, ordinary differential equations, tensors, curvilinear coordinates, integral equations, partial differential equations, and calculus of variations.

APMA 2110. Real Analysis (MATH 2210)
Provides the basis of real analysis which is fundamental to many of the other courses in the program: metric spaces, measure theory, and the theory of integration and differentiation.

APMA 2120. Hilbert Spaces and Their Applications (MATH 2220)
A continuation of APMA 2110: Metric spaces, Banach spaces, Hilbert spaces, the spectrum of bounded operators on Banach and Hilbert spaces, compact operators, applications to integral and differential equations.

APMA 2130. Methods of Applied Mathematics: Partial Differential Equations Solution methods and basic theory for first and second order partial differential equations. Geometric interpretation and solution of linear and nonlinear first order equations by characteristics; formation of caustics and propagation of discontinuities. Classification of second order equations and issues of well-posed problems. Green's functions and maximum principles for elliptic systems. Characteristic methods and discontinuous solutions for hyperbolic systems.

APMA 2140. Methods of Applied Mathematics: Integral Equations
Integral equations. Fredholm and Volterra theory, expansions in orthogonal functions, theory of Hilbert-Schmidt. Singular integral equations, method of Wiener-Hopf. Calculus of variations and direct methods.

APMA 2160. Methods of Applied Mathematics: Asymptotics
Calculus of asymptotic expansions, evaluation of integrals. Solution of linear ordinary differential equations in the complex plane, WKB method, special functions. May be taken concurrently with APMA 2140.

APMA 2170. Functional Analysis and Applications
Topics vary according to interest of instructor and class.

APMA 2190, APMA 2200. Nonlinear Dynamical Systems: Theory and Applications
Basic theory of ordinary differential equations, flows, and maps. Two-dimensional systems. Linear systems. Hamiltonian and integrable systems. Lyapunov functions and stability. Invariant manifolds, including stable, unstable, and center manifolds. Bifurcation theory and normal forms. Nonlinear oscillations and the method of averaging. Chaotic motion, including horseshoe maps and the Melnikov method. Applications in the physical and biological sciences.

APMA 2210. Topics in Differential Equations
A variety of topics in nonlinear dynamics, based in part on the interests of the students, will be covered. Among the possible topics are: bifurcation theory, degree theory, infinite-dimensional systems, delay-differential equations, exponential dichotomies, skew-product flows, and monotone dynamical systems. The prerequisite for the course is a solid (rigorous) grounding in nonlinear dynamics, typically APMA 2190-2200 or equivalent.

APMA 2230, APMA 2240. Partial Differential Equations (MATH 2370, 2380)
The theory of the classical partial differential equations; the method of characteristics and general first order theory. The Fourier transform, the theory of distributions, Sobolev spaces, and techniques of harmonic and functional analysis. More general linear and nonlinear elliptic, hyperbolic, and parabolic equations and properties of their solutions, with examples drawn from physics, differential geometry, and the applied sciences. Semester II concentrates on special topics chosen by the instructor.

APMA 2260. Introduction to Stochastic Control Theory
This course serves as an introduction to the theory of stochastic control and dynamic programming technique. Optimal stopping, total expected (discounted) cost problems, and long-run average cost problems will be discussed in discrete time setting. The last part of the course deals with continuous time deterministic control and game problems. The course requires some familiarity with probability theory.

APMA 2270, APMA 2280. Topics in the Control of Systems
Topics will vary from year to year, but will include optimal control theory, computational methods for control, and algebraic methods in systems science.

APMA 2410. Fluid Dynamics I (ENGN 2810)
Formulation of the basic conservation laws for a viscous, heat conducting, compressible fluid. Molecular basis for thermodynamic and transport properties. Kinematics of vorticity and its transport and diffusion. Introduction to potential flow theory. Viscous flow theory; the application of dimensional analysis and scaling to obtain low and high Reynolds number limits.

APMA 2420. Fluid Dynamics II (ENGN 2820)
A continuation of APMA 2410. Topics include: Low Reynolds number flows, boundary layer theory, wave motion, stability and transition, acoustics, and compressible flows.

APMA 2470, APMA 2480. Topics in Fluid Dynamics
Initial review of topics selected from flow stability, turbulence, turbulent mixing, surface tension effects, and thermal convection. Followed by focused attention on the dynamics of dispersed two-phase flow and complex fluids.

APMA 2550. Numerical Solution of Partial Differential Equations I
Finite difference methods for solving time-depend initial value problems of partial differential equations. Fundamental concepts of consistency, accuracy, stability and convergence of finite difference methods will be covered. Associated well-posedness theory for linear time-dependent PDEs will also be covered. Some knowledge of computer programming expected.

APMA 2560. Numerical Solution of Partial Differential Equations II
Examines the development and analysis of spectral methods for the solution of time-dependent partial differential equations. Topics include key elements of approximation and stability theory for Fourier and polynomial spectral methods as well as attention to temporal integration and numerical aspects. Some knowledge of computer programming expected.

APMA 2570. Numerical Solution of Partial Differential Equations III
We will cover finite element methods for ordinary differential equations and for elliptic, parabolic and hyperbolic partial differential equations. Algorithm development, analysis, and computer implementation issues will be addressed. In particular, we will discuss in depth the discontinuous Galerkin finite element method.

APMA 2580. Computational Fluid Dynamics
An introduction to computational fluid dynamics with emphasis on incompressible flows. Reviews the basic discretization methods (finite differences and finite volumes) following a pedagogical approach from basic operators to the Navier-Stokes equations. Suitable for first-year graduate students, more advanced students, and senior undergraduates. Requirements include three to four computer projects. Material from APMA 1170 and APMA 1180 is appropriate as prerequisite, but no prior knowledge of fluid dynamics is necessary.

APMA 2610. Recent Applications of Probability and Statistics
This is a topics course, covering a selection of modern applications of probability and statistics in the computational, cognitive, engineering, and neural sciences. The course will be rigorous, but the emphasis will be on application. Topics will likely include: Markov chains and their applications to MCMC computing and hidden Markov models; Dependency graphs and Bayesian networks; parameter estimation and the EM algorithm; Kalman and particle filtering; Nonparametric statistics ("learning theory"), including consistency, bias/variance tradeoff, and regularization; the Bayesian approach to nonparametrics, including the Dirichlet and other conjugate priors; principle and independent component analysis; Gibbs distributions, maximum entropy, and their connections to large deviations. Each topic will be introduced with several lectures on the mathematical underpinnings, and concluded with a computer project, carried out by each student individually, demonstrating the mathematics and the utility of the approach. There will be no exams.

APMA 2630, APMA 2640. Theory of Probability (MATH 2630, MATH 2640)
A two-semester course in probability theory. Semester I includes an introduction to probability spaces and random variables, the theory of countable state Markov chains and renewable processes, laws of large numbers and the central limit theorems. Measure theory is first used near the end of the first semester (APMA 2110 may be taken concurrently). Semester II provides a rigorous mathematical foundation to probability theory and covers conditional probabilities and expectations, limit theorems for sums of random variables. martingales, ergodic theory, Brownian motion and an introduction to stochastic process theory.

APMA 2660. Stochastic Processes
Review of the theory of stochastic differential equations and reflected SDEs, and of the ergodic and stability theory of these processes. Introduction to the theory of weak convergence of probability measures and processes. Concentrates on applications to the probabilistic modeling, control, and approximation of modern communications and queuing networks; emphasizes the basic methods, which are fundamental tools throughout applications of probability.

APMA 2670. Mathematical Statistics I
Advanced Statistical Inference. Emphasis on the theoretical aspects of the subject. Frequentist and Bayesian approaches, and their interplay. Topics include: general theory of inference, point and set estimation, hypothesis testing, and modern computational methods (E-M Algorithm, Markov Chain Monte Carlo, Bootstrap). Students should have prior knowledge of probability theory, at the level of APMA 2630 or higher.

APMA 2680. Mathematical Statistics II
This course provides a solid presentation of modern nonparametric statistical methods. Topics include: density estimation, adaptive smoothing, cross-validation, bootstrap, classification and regression trees and their connection to the Huffman code, projection pursuit, the ACE algorithm for time series prediction, support vector machines, and learning theory. The course will provide the mathematical underpinnings, but it will also touch upon some applications in computer vision/speech recognition, and biological, neural, and cognitive sciences. Prerequisite: APMA 2760

APMA 2690, APMA 2700. Topics in Statistics and its Applications
Advanced topics varying from year to year, including: non-parametric methods for density estimation, regression and prediction in time-series; cross-validation and adaptive smoothing techniques; bootstrap; recursive partitioning, projection-pursuit, ACE algorithm; non-parametric classification and clustering; stochastic Metropolis-type simulation and global optimization algorithms; Markov random fields and statistical mechanics; applications to image processing, speech recognition and neural networks.

APMA 2720. Information Theory
Information theory and its relationship with probability, statistics, and data compression. Entropy. The Shannon-McMillan-Breiman theorem. Shannon's source coding theorems. Statistical inference; hypothesis testing; model selection; the minimum description length principle. Information-theoretic proofs of limit theorems in probability: Law of large numbers, central limit theorem, large deviations, Markov chain convergence, Poisson approximation, Hewitt-Savage 0-1 law. Prerequisites: APMA 2630; APMA 1710.

APMA 2810. Seminars in Applied Mathematics Topics Courses

Graduate level seminars in various branches of applied mathematics change from year to year. APMA 2810 Topics Courses are offered during the Fall semester, and APMA 2820 Topics Courses are offered during the Spring semester. The following courses have been offered in past semesters, however for current listings, please see BANNER.

APMA 2810A. Computational Biology
Provides an up-to-date presentation of the main problems and algorithms in bioinformatics. Emphasis is given to statistical/probabilistic methods for various molecular biology tasks, including, comparison of genomes of different species, finding genes and motifs, understanding transcription control mechanisms, analyzing microarray data for gene clustering, and predicting RNA structure.

APMA 2810B. Computational Molecular Biology

Provides an up-to-date presentation of problems and algorithms in bioinformatics, beginning with an introduction to biochemistry and molecular genetics. Topics include: proteins and nucleic acids, the genetic code, the central dogma, the genome, gene expression, metabolic transformations, and experimental methods (gel electrophoresis, X-ray crystallography, NMR). Also, algorithms for DNA sequence alignment, database search tools (BLAST), and DNA sequencing.

APMA 2810C. Elements of High Performance Scientific Computing

APMA 2810D. Elements of High Performance Scientific Computing, II

APMA 2810E. Far Field Boundary Conditions for Hyperbolic Equations

APMA 2810F. Introduction to Non-linear Optics

APMA 2810G. Large Deviations

APMA 2810H. Math of Finance

APMA 2810I. Mathematical Models and Numerical Analysis in Computational Quantum Chemistry.

We shall present on some models in the quantum chemistry field (Thomas Fermi and related, Hartree Fock, Kohn Sham) the basic tools of functional analysis for the study of their solutions. Then some of the discretization methods and iterative algorithms to solve these problems will be presented and analyzed. Some of the open problems that flourish in this field will also be presented all along the lectures.

APMA 2810J. Mathematical Techniques for Neural Modeling

APMA 2810K. Methods of Algebraic Geometry in Control Theory I

Develops the ideas of algebraic geometry in the context of control theory. The first semester examines scalar linear systems and affine algebraic geometry while the second semester addresses multivariable linear systems and projective algebraic geometry.

APMA 2810L. Numerical Solution of Hyperbolic PDE’s

APMA 2810M. Some Topics in Kinetic Theory

Nonlinear instabilities as well as boundary effects in a collisionless plasmas; Stable galaxy configurations; A nonlinear energy method in the Boltzmann theory will also be introduced. Self-contained solutions to specific concrete problems. Focus on ideas but not on technical aspects. Open problems and possible future research directions will then be discussed so that students can gain a broader perspective. Prerequisite: One semester of PDE (graduate level) is required.

APMA 2810N. Topics in Nonlinear PDEs

Aspects of the theory on nonlinear evolution equations, which includes kinetic theory, nonlinear wave equations, variational problems, and dynamical stability.

APMA 2810O. Stochastic Differential Equations
This course develops the theory and some applications of stochastic differential equations. Topics include: stochastic integral with respect to Brownian motion, existence and uniqueness for solutions of SDEs, Markov property of solutions, sample path properties, Girsanov's Theorem, weak existence and uniqueness, and connections with partial differential equations. Possible additional topics include stochastic stability, reflected diffusions, numerical approximation, and stochastic control. Prerequisite: APMA 2630, 2640

APMA 2810P. Perturbation Methods

Basic concepts of asymptotic approximations with examples with examples such as evaluation of integrals and functions. Regular and singular perturbation problems for differential equations arising in fluid mechanics, wave propagation or nonlinear oscillators. Methods include matched asymptotic expansions and multiple scales. Methods and results will be discussed in the context of applications to physical problems.

APMA 2810Q. Discontinuous Galerkin Methods

In this seminar course we will cover the algorithm formulation, stability analysis and error estimates, and implementation and applications of discontinuous Galerkin finite element methods for solving hyperbolic conservation laws, convection diffusion equations, dispersive wave equations, and other linear and nonlinear partial differential equations. Prerequisite: APMA 2550.

APMA 2810R. Computational Biology Methods for Gene/Protein Networks and Structural Proteomics

The course presents computational and statistical methods for gene and protein networks and structural proteomics; it emphasizes: (1) Probablistic models for gene regulatory networks via microarray, chromatin immune-precipitation, and cis-regulatory data; (2) Signal transduction pathways via tandem mass spectrometry data; (3) Molecular Modeling forligand-receptor coupling and docking. The course is recommended for graduate students.

APMA 2810S. Topics in Control

APMA 2810T. Nonlinear Partial Differential Equations

This course introduces techniques useful for solving many nonlinear partial differential equations, with emphasis on elliptic problems. PDE from a variety of applications will be discussed. Contact the instructor about prerequisites.

APMA 2810U. Topics in Differential Equations

APMA 2810V. Topics in Partial Differential Equations
The course will cover an introduction of the L_p theory of second order elliptic and parabolic equations, finite difference approximations of elliptic and parabolic equations, and some recent developments in the Navier-Stokes equations and quasi-geotropic equations. Some knowledge of real analysis will be expected.

APMA 2810W. Advanced Topics in High Order Numerical Methods for Convection Dominated Problems
This is an advanced seminar course. We will cover several topics in high order numerical methods for convection dominated problems, including methods for solving Boltzmann type equations, methods for solving unsteady and steady Hamilton-Jacobi equations, and methods for solving moment models in semi-conductor device simulations. Prerequisite: APMA 2550 or equivalent knowledge of numerical analysis.

APMA 2810X. Introduction to the Theory of Large Deviations
The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's and Cramer's Theorems). We then will cover large deviations for diffusion processes and the Wentsel-Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman equations; Monte Carlo methods. Prerequisites: APMA 2630 and 2640.

APMA 2810Y. Discrete high-D Inferences in Genomics
Genomics is revolutionizing biology and biomedicine and generated a mass of clearly relevant high-D data along with many important high-D discreet inference problems. Topics: special characteristics of discrete high-D inference including Bayesian posterior inference; point estimation; interval estimation; hypothesis tests; model selection; and statistical decision theory.

APMA 2811A. Directed Methods in Control and System Theory. Various general techniques have been developed for control and system problems. Many of the methods are indirect. For example, control problems are reduced to a problem involving a differential equation (such as the partial differential equation of Dynamic Programming) or to a system of differential equations (such as the canonical system of the Maximum Principle). Since these indirect methods are not always effective alternative approaches are necessary. In particular, direct methods are of interest. We deal with two general classes, namely: 1.) Integration Methods; and, 2.) Representation Methods. Integration methods deal with the integration of function space differential equations. Perhaps the most familiar is the so-called Gradient Method or curve of steepest descent approach. Representation methods utilize approximation in function spaces and include both deterministic and stochastic finite element methods. Our concentration will be on the theoretical development and less on specific numerical procedures. The material on representation methods for Levy processes is new.

APMA 2811G. Topics in Averaging and Metastability with Applications
Topics that will be covered include: the averaging principle for stochastic dynamical systems and in particular for Hamiltonian systems; metastability and stochastic resonance. We will also discuss applications in class and in homework problems. In particular we will consider metastability issues arising in chemistry and biology, e.g. in the dynamical behavior of proteins. The course will be largely self contained, but a course in graduate probability theory and/or stochastic calculus will definitely help.

APMA 2820. Seminar in Applied Mathematics Topics Courses

Graduate level seminars in various branches of applied mathematics change from year to year. The below mentioned courses are typically offered during the Spring semester. For current listings, please see BANNER.

APMA 2820A. A Tutorial on Particle Methods

APMA 2820B. Advanced Topics in Information Theory

The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's and Cramer's Theorems). We then will cover large deviations for diffusion processes and the Wentsel-Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman equations; Monte Carlo methods. Prerequisites: AM 263 and 264.

APMA 2820C. Computational Electromagnetics

APMA 2820D. Conventional, Real and Quantum Computing with Applications to Factoring and Root Finding

APMA 2820E. Geophysical Fluid Dynamics

APMA 2820F. Information Theory and Networks

APMA 2820G. Information Theory, Statistics and Probability

APMA 2820H. Kinetic Theory

We will focus on two main topics in mathematical study of the kinetic theory: (1) The new goal method to study the trend to Maxwellians; (2) various hydrodynamical (fluids) limits to Euler and Navier-Stokes equations. Main emphasis will be on the ideas behind proofs, but not on technical details.

APMA 2820I. Multiscale Methods and Computer Vision

Course will address some basic multiscale computational methods such as: multigrid solvers for physical systems, including both geometric and algebraic multigrid, fast integral transforms of various kinds (including a fast Radon transform), and fast inverse integral transforms. Basic problems in computer vision such as global contour detection and their completion over gaps, image segmentation for textural images and perceptual grouping tasks in general will be explained in more details.

APMA 2820J. Numerical Linear Algebra
Solving large systems of linear equations: The course will use the text of Treften and Bao that includes all the modern concepts of solving linear questions.

APMA 2820K. Numerical Solution of Ordinary Differential Equations

We discuss the construction and general theory of multistep and multistage methods for numerically solving systems of ODE's, including stiff and nonlinear problems. Different notions to stability and error estimation and control. As time permits we shall discuss more advanced topics such as order reduction, general linear and additive methods, symplectic methods, and methods for DAE. Prerequisites: AM 219 and AM 255 or equivalent. Some programming experience is expected.

APMA 2820L. Random Processes in Mechanics

APMA 2820M. Singularities in Elliptic Problems and their Treatment by High-Order Finite Element Methods

Singular solutions for elliptic problems (elasticity and heat transfer) are discussed. These may arise around corners in 2-D and along edges and vertices in 3-D domains. Derivation of singular solutions, characterized by eigenpairs and generalized stress/flux intensity factors (GSIF/GFIFs) are a major engineering importance (because of failure initiation and propagation). High-order FE methods are introduced, and special algorithms for extracting eigenpairs and GSIF/GFIFs are studied (Steklov, dual-function, ERR method, and others).

APMA 2820O. The Mathematics of Shape with Applications to Computer Vision

Methods of representing shape, the geometry of the space of shapes, warping and matching of shapes, and some applications to problems in computer vision and medical imaging. Prerequisite: See instructor for prerequisites.

APMA 2820P. Foundations in Statistical Inference in Molecular Biology
In molecular biology, inferences in high dimensions with missing data are common. A conceptual framework for Bayesian and frequentist inferences in this setting include: sequence alignment, RNA secondary structure prediction, database search, and tiled arrays. Statistical topics: parameter estimation, hypothesis testing, recursions, and characterization of posterior spaces. This is a core course in proposed Ph.D. program in computational molecular biology.

APMA 2820Q. Topics in Kinetic Theory

This course will introduce current mathematical study for Boltzmann equation and Vlasov equation. We will study large time behavior and hydrodynamic limits for Boltzmann theory and instabilities in the Vlasov theory. Graduate PDE course is required.

APMA 2820R. Structure Theory of Control Systems
The course deals with the following problems: Given a family of control systems S and a family of control systems S', when does there exist an appropriate embedding of S into S'? Most of the course with deal with families of linear control systems. Knowledge of control theory and mathematical sophistication are required.

APMA 2820S. Topics in Differential Equations
A sequel to APMA 2210 concentrating on similar material.

APMA 2820T. Foundations in Statistical Inference in Molecular Biology
In molecular biology, inferences in high dimensions with missing data are common. A conceptual framework for Bayesian and frequentist inferences in this setting including: sequence alignment. RNA secondary structure prediction, database search, and tiled arrays. Statistical topics: parameter estimation, hypothesis testing, recursions, and characterization of posterior spaces. Core course in proposed PhD program in computational molecular biology.

APMA 2820U. Structure Theory of Control Systems
The course deals with the following problems: given a family of control systems S and a family of control systems S', when does there exist an appropriate embedding of S into S'? Most of the course will deal with the families of linear control systems. Knowledge of control theory and mathematical sophistication are required.

APMA 2820V. Progress in the Theory of Shock Waves
The course will begin with a self-contained introduction to the theory of "hyperbolic conservation laws," that is quasilinear first order systems of partial differential equations whose solutions spontaneously develop singularities that propagate as shock waves. Then a number of recent developments will be discussed. The aim is to familiarize the students with the current status of the theory as well as with the ever expanding areas of applications of the subject.

APMA 2820W. An introduction to the Theory of Large Deviations
The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's and Cramer's Theorems). We then will cover large deviations for diffusion processes and the Wentsel-Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman equations; Monte Carlo methods. Prerequisites: APMA 2630 and APMA 2640.

APMA 2820X. Boundary Conditions for Hyperbolic Systems: Numerical and Far Field

APMA 2820Y. Approaches to Problem Solving in Applied Mathematics

APMA 2820Z. Topics in Discontinuous Galerkin Methods
In molecular biology, inferences in high dimensions with missing data are common. A conceptual framework for Bayesian and frequentist inferences in this setting including: sequence alignment. RNA secondary structure prediction, database search, and tiled arrays. Statistical topics: parameter estimation, hypothesis testing, recursions, and characterization of posterior spaces. Core course in proposed PhD program in computational molecular biology.

APMA 2821A. Parallel Scientific Computing: Algorithms and Tools

APMA 2821D. Random Processes and Random Variables.

APMA 2970. Preliminary Examination Preparation

APMA 2980. Research in Applied Mathematics

APMA 2990. Thesis Preparation